15
15

Sep 18, 2013
09/13

by
Mikhail Khovanov

######
15

######
0

######
0

The sequence of rings $H^n, n\ge 0,$ introduced in math.QA/0103190, controls categorification of the quantum sl(2) invariant of tangles. We prove that the center of $H^n$ is isomorphic to the cohomology ring of the (n,n) Springer variety and show that the braid group action in the derived category of $H^n$-modules descends to the Springer action of the symmetric group.

Source: http://arxiv.org/abs/math/0202110v1

17
17

Sep 21, 2013
09/13

by
Mikhail Khovanov

######
17

######
0

######
0

A diagrammatic presentation of functors and natural transformations and the virtues of biadjointness are discussed. We then review a graphical description of the category of Soergel bimodules and a diagrammatic categorification of positive halves of quantum groups. These notes are a write-up of Takagi lectures given by the author in Hokkaido University in June 2009.

Source: http://arxiv.org/abs/1008.5084v1

15
15

Sep 23, 2013
09/13

by
Mikhail Khovanov

######
15

######
0

######
0

We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triply-graded link homology and relate it to Kazhdan-Lusztig theory.

Source: http://arxiv.org/abs/math/0510265v3

7
7.0

Sep 19, 2013
09/13

by
Mikhail Khovanov

######
7

######
0

######
0

The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.

Source: http://arxiv.org/abs/math/0302060v1

28
28

Sep 23, 2013
09/13

by
Mikhail Khovanov

######
28

######
0

######
0

We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.

Source: http://arxiv.org/abs/math/9908171v2

11
11

Sep 24, 2013
09/13

by
Mikhail Khovanov

######
11

######
0

######
0

We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle. When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.

Source: http://arxiv.org/abs/math/0103190v2

18
18

Sep 22, 2013
09/13

by
Mikhail Khovanov

######
18

######
0

######
0

Cohomology theory of links, introduced by the author, is combinatorial. Dror Bar-Natan recently wrote a program that found ranks of cohomology groups of all prime knots with up to 11 crossings. His surprising experimental data is discussed in this note.

Source: http://arxiv.org/abs/math/0201306v1

13
13

Sep 18, 2013
09/13

by
Mikhail Khovanov

######
13

######
0

######
0

This is a short survey of algebro-combinatorial link homology theories which have the Jones polynomial and other link polynomials as their Euler characteristics.

Source: http://arxiv.org/abs/math/0605339v2

9
9.0

Sep 18, 2013
09/13

by
Mikhail Khovanov

######
9

######
0

######
0

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

Source: http://arxiv.org/abs/math/0509083v2

42
42

Sep 17, 2013
09/13

by
Mikhail Khovanov

######
42

######
0

######
0

We prove that the construction of our previous paper math.QA/0103190 yields an invariant of tangle cobordisms.

Source: http://arxiv.org/abs/math/0207264v1

19
19

Sep 19, 2013
09/13

by
Mikhail Khovanov

######
19

######
0

######
0

We define a bigraded homology theory whose Euler characteristic is the quantum sl(3) link invariant.

Source: http://arxiv.org/abs/math/0304375v2

12
12

Jul 20, 2013
07/13

by
Mikhail Khovanov

######
12

######
0

######
0

We describe a collection of differential graded rings that categorify weight spaces of the positive half of the quantized universal enveloping algebra of the Lie superalgebra gl(1|2).

Source: http://arxiv.org/abs/1007.3517v2

22
22

Sep 23, 2013
09/13

by
Mikhail Khovanov

######
22

######
0

######
0

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

Source: http://arxiv.org/abs/math/0411447v2

18
18

Jul 20, 2013
07/13

by
Mikhail Khovanov

######
18

######
0

######
0

We show that induction and restriction functors for inclusions of nilCoxeter algebras provide a categorical realization of the algebra of polynomial differential operators in one variable.

Source: http://arxiv.org/abs/math/9906166v1

7
7.0

Sep 19, 2013
09/13

by
Mikhail Khovanov

######
7

######
0

######
0

A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of vector spaces of morphisms between products of generating objects in this category.

Source: http://arxiv.org/abs/1009.3295v1

12
12

Sep 24, 2013
09/13

by
Pavel Etingof; Mikhail Khovanov

######
12

######
0

######
0

In this note we illustrate by a few examples the general principle: interesting algebras and representations defined over Z_+ come from category theory, and are best understood when their categorical origination has been discovered. We show that indecomposable Z_+-representations of the character ring of SU(2) satisfying certain conditions correspond to affine and infinite Dynkin diagrams with loops. We also show that irreducible Z_+-representations of the Verlinde algebra (the character ring...

Source: http://arxiv.org/abs/hep-th/9408078v2

6
6.0

Sep 21, 2013
09/13

by
Ben Elias; Mikhail Khovanov

######
6

######
0

######
0

The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.

Source: http://arxiv.org/abs/0902.4700v1

4
4.0

Sep 18, 2013
09/13

by
Mikhail Khovanov; Lev Rozansky

######
4

######
0

######
0

To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.

Source: http://arxiv.org/abs/math/0505056v2

5
5.0

Sep 18, 2013
09/13

by
Mikhail Khovanov; Paul Seidel

######
5

######
0

######
0

We consider the derived categories of modules over a certain family A_m of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type A_m. We show that each of these two rather different objects encodes the topology of curves on an (m+1)-punctured disc. We prove that the braid group B_{m+1} acts faithfully on the derived category of A_m-modules and that it injects into the symplectic mapping class group...

Source: http://arxiv.org/abs/math/0006056v2

13
13

Jul 20, 2013
07/13

by
Mikhail Khovanov; Richard Thomas

######
13

######
0

######
0

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

Source: http://arxiv.org/abs/math/0609335v2

13
13

Sep 19, 2013
09/13

by
Yanfeng Chen; Mikhail Khovanov

######
13

######
0

######
0

We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).

Source: http://arxiv.org/abs/math/0610054v3

7
7.0

Sep 19, 2013
09/13

by
Mikhail Khovanov; Lev Rozansky

######
7

######
0

######
0

We suggest a categorification procedure for the SO(2N) one-variable specialization of the two-variable Kauffman polynomial. The construction has many similarities with the HOMFLYPT categorification: a planar graph formula for the polynomial is converted into a complex of graded vector spaces, each of them being the homology of a Z_2 graded differential vector space associated to a graph and constructed using matrix factorizations. This time, however, the elementary matrix factorizations are not...

Source: http://arxiv.org/abs/math/0701333v1

38
38

Jul 20, 2013
07/13

by
Mikhail Khovanov; Greg Kuperberg

######
38

######
0

######
0

We compare two natural bases for the invariant space of a tensor product of irreducible representations of A_2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A_2 spider. The other basis is the dual canonical basis, the dual of the basis defined by Lusztig and Kashiwara. For sl(2) or A_1, the web bases have been discovered many times and were recently shown to be dual canonical by Frenkel and Khovanov. We prove that for sl(3), the two bases eventually...

Source: http://arxiv.org/abs/q-alg/9712046v2

26
26

Sep 18, 2013
09/13

by
Mikhail Khovanov; Lev Rozansky

######
26

######
0

######
0

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.

Source: http://arxiv.org/abs/math/0401268v2

7
7.0

Sep 22, 2013
09/13

by
Mikhail Khovanov; Radmila Sazdanovic

######
7

######
0

######
0

We develop a diagrammatic categorification of the polynomial ring $Z[x]$. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to $x^n$ and standard modules to $(x-1)^n$ in the Grothendieck ring.

Source: http://arxiv.org/abs/1101.0293v1

113
113

Jul 20, 2013
07/13

by
Marta Asaeda; Mikhail Khovanov

######
113

######
0

######
0

This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah.

Source: http://arxiv.org/abs/0804.1279v1

6
6.0

Sep 20, 2013
09/13

by
Ruth Stella Huerfano; Mikhail Khovanov

######
6

######
0

######
0

We categorify representations of quantum sl(n) whose highest weight is twice a fundamental weight.

Source: http://arxiv.org/abs/math/0204333v1

7
7.0

Sep 18, 2013
09/13

by
Ruth Stella Huerfano; Mikhail Khovanov

######
7

######
0

######
0

We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of C. The category C is the direct sum of a semisimple category and the category of modules over a certain algebra A, associated to a Dynkin diagram. In the second half of the paper we show how these algebras appear in the...

Source: http://arxiv.org/abs/math/0002060v2

18
18

Jul 20, 2013
07/13

by
Alexander P. Ellis; Mikhail Khovanov

######
18

######
0

######
0

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q=-1 case leads to a Z-graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting we describe counterparts of the elementary and complete symmetric functions, power sums, Schur functions, and combinatorial interpretations of associated change of basis relations.

Source: http://arxiv.org/abs/1107.5610v1

9
9.0

Sep 18, 2013
09/13

by
Mikhail Khovanov; Aaron D. Lauda

######
9

######
0

######
0

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph.

Source: http://arxiv.org/abs/0803.4121v2

60
60

Jul 20, 2013
07/13

by
Mikhail Khovanov; Aaron D. Lauda

######
60

######
0

######
0

We categorify one-half of the quantum group associated to an arbitrary Cartan datum.

Source: http://arxiv.org/abs/0804.2080v1

38
38

Jul 24, 2013
07/13

by
Mikhail Khovanov; Aaron D. Lauda

######
38

######
0

######
0

We categorify the idempotented form of quantum sl(n).

Source: http://arxiv.org/abs/0807.3250v1

5
5.0

Sep 19, 2013
09/13

by
Igor Frenkel; Mikhail Khovanov; Alexander Kirillov Jr

######
5

######
0

######
0

In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of the quantum group $U_q sl_k$. We also use known results about canonical bases for $U_q sl_2$ to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmanians), due to...

Source: http://arxiv.org/abs/q-alg/9709042v1

10
10.0

Sep 23, 2013
09/13

by
Igor Frenkel; Mikhail Khovanov; Catharina Stroppel

######
10

######
0

######
0

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra gl(n). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed. We also give a...

Source: http://arxiv.org/abs/math/0511467v1

7
7.0

Sep 21, 2013
09/13

by
Anna Beliakova; Mikhail Khovanov; Aaron D. Lauda

######
7

######
0

######
0

We categorify the Casimir element of the idempotented form of quantum sl(2).

Source: http://arxiv.org/abs/1008.0370v2

10
10.0

Sep 21, 2013
09/13

by
Igor Frenkel; Mikhail Khovanov; Olivier Schiffmann

######
10

######
0

######
0

We describe points on Nakajima varieties and Weyl group actions on them via complexes of semisimple and projective modules over certain finite-dimensional algebras.

Source: http://arxiv.org/abs/math/0311485v1

12
12

Sep 20, 2013
09/13

by
Mikhail Khovanov; Volodymyr Mazorchuk; Catharina Stroppel

######
12

######
0

######
0

We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_n(\mathbb{C})$.

Source: http://arxiv.org/abs/math/0607630v1

59
59

Sep 18, 2013
09/13

by
Joseph Bernstein; Igor Frenkel; Mikhail Khovanov

######
59

######
0

######
0

We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is...

Source: http://arxiv.org/abs/math/0002087v1

9
9.0

Sep 22, 2013
09/13

by
Mikhail Khovanov; Volodymyr Mazorchuk; Catharina Stroppel

######
9

######
0

######
0

This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra sl(n). The review is intended to give non-experts in representation theory who...

Source: http://arxiv.org/abs/math/0702746v2

9
9.0

Sep 23, 2013
09/13

by
Alexander P. Ellis; Mikhail Khovanov; Aaron D. Lauda

######
9

######
0

######
0

We introduce an odd version of the nilHecke algebra and develop an odd analogue of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogues of the cohomology rings of Grassmannians. Like their even counterparts, odd nilHecke algebras...

Source: http://arxiv.org/abs/1111.1320v1

9
9.0

Sep 23, 2013
09/13

by
Mikhail Khovanov; Aaron D. Lauda; Marco Mackaay; Marko Stosic

######
9

######
0

######
0

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection...

Source: http://arxiv.org/abs/1006.2866v1